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i recently read a book about Godel's theorm. book was discussing the problem of consistancy in formal systems. it talked about two general ways to insure consystency: a method of modeling, when one takes a new system's postulates and converts into valid theorems of already known system.(say, you come up with a model within euclidian geometry and try to see that a noneuclidian geometry's postulates can be converted into valid theorems of euclidian geometry) then author points out the problem with such aproach, namely you still have to show the consistensy of the already known system, in the case above of euclidian geometry. second aproach, he talks about, is to completely formalize the system.

anyways, in both methods i coud not see how they would guarantee consistensy. could you please help me out and tell me if there's really any logical way to prove that a given formal system will be consistent, for example in number theory there's no way one can prove nonexistanse of the highest prime number. and i do not mean it has been proved such number does not exist(i know how to prove it) but that there is no way from infinite number of theorems within the system one can show they will not combine in such a way that conclusion will be highest prime number exist.

please, give more laymen's explanation since i am not professional logician/mathematician.

thank you all

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# Consistency of formal system

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